1. Introduction to Robotics
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2. General Course Information
The course introduces the basic concepts of robotic manipulators and autonomous systems. After a review of some fundamental mathematics the course examines the mechanics and dynamics of robot arms, mobile robots, their sensors and algorithms for controlling them.
two robotic arms
everything in Matlab (and some Java)
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3. A150 Robotic Arm link 3 link 2 Symbolic Representation of Manipulators
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4. Kinematics
the study of motion that ignores the forces that cause the motion
“geometry of motion”
interested in position, velocity, acceleration, etc. of the various links of the manipulator
e.g., where is the gripper relative to the base of the manipulator? what direction is it pointing in?
described using rigid transformations of the links
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5. Kinematics forward kinematics: inverse kinematics:
given the link lengths and joint angles compute the position and orientation of the gripper relative to the base
for a serial manipulator there is only one solution
inverse kinematics:
given the position (and possibly the orientation) of the gripper and the dimensions of the links, what are the joint variables?
for a serial manipulator there is often more than one mathematical solution
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6. Wheeled Mobile Robots
robot can have one or more wheels that can provide
steering (directional control)
power (exert a force against the ground)
an ideal wheel is
perfectly round (perimeter 2πr)
moves in the direction perpendicular to its axis
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7. Wheel
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8. Deviations from Ideal
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9. Differential Drive
two independently driven wheels mounted on a common axis
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10. Forward Kinematics
for a robot starting with pose [ ]T moving with velocity V(t) in a direction θ(t) :
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11. Sensitivity to Wheel Velocity
σ = 0.05
σ = 0.01
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12. Localization using Landmarks: RoboSoccer
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13. Maps
goal *
start
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14. Path Finding
goal
start
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15. Localization
goal
start
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16. EKF SLAM Application www.probabilistic-robotics.org
[MIT B21, courtesy by John Leonard]

17. EKF SLAM Application raw odometry estimated trajectory
[courtesy by John Leonard]

18. Introduction to manipulator kinematics
Day 02
Introduction to manipulator kinematics
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19. Robotic Manipulators a robotic manipulator is a kinematic chain
i.e. an assembly of pairs of rigid bodies that can move respect to one another via a mechanical constraint
the rigid bodies are called links
the mechanical constraints are called joints
Symbolic Representation of Manipulators
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20. A150 Robotic Arm link 3 link 2 Symbolic Representation of Manipulators
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21. Joints most manipulator joints are one of two types
revolute (or rotary)
like a hinge
allows relative rotation about a fixed axis between two links
axis of rotation is the z axis by convention
prismatic (or linear)
like a piston
allows relative translation along a fixed axis between two links
axis of translation is the z axis by convention
our convention: joint i connects link i – 1 to link i
when joint i is actuated, link i moves
Symbolic Representation of Manipulators
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22. Joint Variables
revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable
qi : joint variable for joint i
revolute
qi = qi : angle of rotation of link i relative to link i – 1
prismatic
qi = di : displacement of link i relative to link i – 1
Symbolic Representation of Manipulators
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23. Revolute Joint Variable
qi = qi : angle of rotation of link i relative to link i – 1
link i qi
link i – 1
Symbolic Representation of Manipulators
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24. Prismatic Joint Variable
qi = di : displacement of link i relative to link i – 1
link i – 1
link i di
Symbolic Representation of Manipulators
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25. Common Manipulator Arrangments
most industrial manipulators have six or fewer joints
the first three joints are the arm
the remaining joints are the wrist
it is common to describe such manipulators using the joints of the arm
R: revolute joint
P: prismatic joint
Common Manipulator Arrangements
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26. Articulated Manipulator
RRR (first three joints are all revolute)
joint axes
z0 : waist
z1 : shoulder (perpendicular to z0)
z2 : elbow (parallel to z1) z0 z1 z2 q2 q3
shoulder
forearm
elbow q1
waist
Common Manipulator Arrangements
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27. Spherical Manipulator
RRP
Stanford arm z0 z1 d3 q2
shoulder z2 q1
waist
Common Manipulator Arrangements
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28. SCARA Manipulator
RRP
Selective Compliant Articulated Robot for Assembly z1 z2 z0 q2 d3 q1
Common Manipulator Arrangements
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29. Forward Kinematics
given the joint variables and dimensions of the links what is the position and orientation of the end effector? a2 q2 a1 q1
Forward Kinematics
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30. Forward Kinematics choose the base coordinate frame of the robot
we want (x, y) to be expressed in this frame
(x, y) ? a2 q2 y0 a1 q1 x0
Forward Kinematics
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31. Forward Kinematics
notice that link 1 moves in a circle centered on the base frame origin
(x, y) ? a2 q2 y0 a1 q1
( a1 cos q1 , a1 sin q1 ) x0
Forward Kinematics
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32. Forward Kinematics
choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame
(x, y) ? y1 a2 q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0
Forward Kinematics
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33. Forward Kinematics
notice that link 2 moves in a circle centered on frame 1
(x, y) ? y1 a2
( a2 cos (q1 + q2),
a2 sin (q1 + q2) ) q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0
Forward Kinematics
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34. Forward Kinematics
because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame
(a1 cos q1 + a2 cos (q1 + q2),
a1 sin q1 + a2 sin (q1 + q2) ) y1 a2
( a2 cos (q1 + q2),
a2 sin (q1 + q2) ) q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0
Forward Kinematics
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35. Forward Kinematics
we also want the orientation of frame 2 with respect to the base frame
x2 and y2 expressed in terms of x0 and y0 y2 x2 a2 q2 y0 q1 a1 q1 x0
Forward Kinematics
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36. Forward Kinematics without proof I claim: x2 = (cos (q1 + q2),
sin (q1 + q2) ) y2 x2
y2 = (-sin (q1 + q2),
cos (q1 + q2) ) a2 q2 y0 q1 a1 q1 x0
Forward Kinematics
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37. Inverse Kinematics
given the position (and possibly the orientation) of the end effector, and the dimensions of the links, what are the joint variables? y2 x2
(x, y) a2
q2 ? y0 a1
q1 ? x0
Inverse Kinematics
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38. Inverse Kinematics
harder than forward kinematics because there is often more than one possible solution
(x, y) a2 y0 a1 x0
Inverse Kinematics
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39. Inverse Kinematics law of cosines (x, y) a2 b q2 ? y0 a1 x0
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40. Inverse Kinematics and we have the trigonometric identity therefore,
We could take the inverse cosine, but this gives only one of the two solutions.
Inverse Kinematics
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41. Inverse Kinematics Instead, use the two trigonometric identities:
to obtain
which yields both solutions for q2 . In many programming languages you would use the
four quadrant inverse tangent function atan2
c2 = (x*x + y*y – a1*a1 – a2*a2) / (2*a1*a2);
s2 = sqrt(1 – c2*c2);
theta21 = atan2(s2, c2);
theta22 = atan2(-s2, c2);
Inverse Kinematics
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42. Inverse Kinematics Exercise for the student: show that
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43. Day 03
Spatial Descriptions
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44. Points and Vectors point : a location in space
vector : magnitude (length) and direction between two points
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45. Coordinate Frames
choosing a frame (a point and two perpendicular vectors of unit length) allows us to assign coordinates
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46. Coordinate Frames
the coordinates change depending on the choice of frame
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47. Dot Product
the dot product of two vectors
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48. Translation
suppose we are given o1 expressed in {0}
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49. Translation 1
the location of {1} expressed in {0}
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50. Translation 1
the translation vector can be interpreted as the location of frame {j} expressed in frame {i}
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51. Translation 2 p1 expressed in {0} a point expressed in frame {1}
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52. Translation 2
the translation vector can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}
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53. Translation 3
q0 expressed in {0}
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54. Translation 3
the translation vector can be interpreted as an operator that takes a point and moves it to a new point in the same frame
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55. Rotation suppose that frame {1} is rotated relative to frame {0}
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56. Rotation 1
the orientation of frame {1} expressed in {0}
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57. Rotation 1
the rotation matrix can be interpreted as the orientation of frame {j} expressed in frame {i}
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58. Rotation 2
p1 expressed in {0}
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59. Rotation 2
the rotation matrix can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}
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60. Rotation 3
q0 expressed in {0}
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61. Rotation 3
the rotation matrix can be interpreted as an operator that takes a point and moves it to a new point in the same frame
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62. Properties of Rotation Matrices
RT = R-1
the columns of R are mutually orthogonal
each column of R is a unit vector
det R = 1 (the determinant is equal to 1)
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63. Rotation and Translation
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64. Rotations in 3D
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65. Day 04
Rotations
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66. Properties of Rotation Matrices
RT = R-1
the columns of R are mutually orthogonal
each column of R is a unit vector
det R = 1 (the determinant is equal to 1)
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67. Rotations in 3D
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68. Rotation About z-axis
+'ve rotation
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69. Rotation About x-axis
+'ve rotation
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70. Rotation About y-axis
+'ve rotation
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71. Relative Orientation Example
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72. Successive Rotations in Moving Frames
Suppose you perform a rotation in frame {0} to obtain {1}.
Then you perform a rotation in frame {1} to obtain {2}.
What is the orientation of {2} relative to {0}?
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73. Successive Rotations in a Fixed Frame
Suppose you perform a rotation in frame {0} to obtain {1}.
Then you rotate {1} in frame {0} to obtain {2}.
What is the orientation of {2} relative to {0}?
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74. Composition of Rotations
Given a fixed frame {0} and a current frame {1} and
if {2} is obtained by a rotation R in the current frame {1} then use postmulitplication to obtain:
Given a fixed frame {0} and a frame {1} and
if {2} is obtained by a rotation R in the fixed frame {0} then use premultiplication to obtain:
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75. Rotation About a Unit Axis
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76. Rigid Body Transformations
Day 05
Rigid Body Transformations
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77. Homogeneous Representation
translation represented by a vector d
vector addition
rotation represented by a matrix R
matrix-matrix and matrix-vector multiplication
convenient to have a uniform representation of translation and rotation
obviously vector addition will not work for rotation
can we use matrix multiplication to represent translation?
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78. Homogeneous Representation
consider moving a point p by a translation vector d
not possible as matrix-vector multiplication always leaves the origin unchanged
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79. Homogeneous Representation
consider an augmented vector ph and an augmented matrix D
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80. Homogeneous Representation
the augmented form of a rotation matrix R3x3
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81. Rigid Body Transformations in 3D
{1}
{0}
{2}
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82. Rigid Body Transformations in 3D
suppose {1} is a rotated and translated relative to {0}
what is the pose (the orientation and position) of {1} expressed in {0} ?
{1}
{0}
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83. Rigid Body Transformations in 3D
suppose we use the moving frame interpretation (postmultiply transformation matrices)
translate in {0} to get {0’}
and then rotate in {0’} to get {1}
{0’}
{0}
{0’}
{1}
{0}
Step 1
Step 2
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84. Rigid Body Transformations in 3D
suppose we use the fixed frame interpretation (premultiply transformation matrices)
rotate in {0} to get {0’}
and then translate in {0} in to get {1}
{0}
{0’}
{1}
{0}
Step 1
{0’}
Step 2
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85. Rigid Body Transformations in 3D
both interpretations yield the same transformation
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86. Homogeneous Representation
every rigid-body transformation can be represented as a rotation followed by a translation in the same frame
as a 4x4 matrix
where R is a 3x3 rotation matrix and d is a 3x1 translation vector
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87. Homogeneous Representation
in some frame i
points
vectors
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88. Inverse Transformation
the inverse of a transformation undoes the original transformation if
then
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89. Day 06
Forward Kinematics
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90. Transform Equations
{2}
{1}
{0}
{4}
{3}
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91. Transform Equations
give expressions for:
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92. Transform Equations
{1}
{0}
{3}
{2}
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93. Transform Equations
how can you find
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94. Links and Joints n joints, n + 1 links link 0 is fixed (the base)
joint i connects link i – 1 to link i
link i moves when joint i is actuated
joint n
joint 3
joint 4
joint n-1
link 3
joint 1
link 2
link n-1
link n
link 1
joint 2
link 0
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95. Forward Kinematics
given the joint variables and dimensions of the links what is the position and orientation of the end effector?
p0 ? a2 q2 y0 a1 q1 x0
Forward Kinematics
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96. Forward Kinematics
because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame
(a1 cos q1 + a2 cos (q1 + q2),
a1 sin q1 + a2 sin (q1 + q2) ) y1 a2
( a2 cos (q1 + q2),
a2 sin (q1 + q2) ) q2 y0 q1 a1 x1 q1
( a1 cos q1 , a1 sin q1 ) x0
Forward Kinematics
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97. Forward Kinematics from Day 02 p0 = (a1 cos q1 + a2 cos (q1 + q2),
a1 sin q1 + a2 sin (q1 + q2) ) y2 x2
x2 = (cos (q1 + q2),
sin (q1 + q2) ) a2
y2 = (-sin (q1 + q2),
cos (q1 + q2) ) q2 y0 q1 a1 q1 x0
Forward Kinematics
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98. Frames y2 x2 y1 a2 q2 y0 x1 a1 q1 x0
Forward Kinematics
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99. Forward Kinematics
using transformation matrices
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100. Day 07
Denavit-Hartenberg
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101. Links and Joints n joints, n + 1 links link 0 is fixed (the base)
joint i connects link i – 1 to link i
link i moves when joint i is actuated
joint n
joint 3
joint 4
joint n-1
link 3
joint 1
link 2
link n-1
link n
link 1
joint 2
link 0
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102. Forward Kinematics attach a frame {i} to link i
all points on link i are constant when expressed in {i}
if joint i is actuated then frame {i} moves relative to frame {i - 1}
motion is described by the rigid transformation
the state of joint i is a function of its joint variable qi (i.e., is a function of qi)
this makes it easy to find the last frame with respect to the base frame
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103. Forward Kinematics more generally
the forward kinematics problem has been reduced to matrix multiplication
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104. Forward Kinematics
Denavit J and Hartenberg RS, “A kinematic notation for lower- pair mechanisms based on matrices.” Trans ASME J. Appl. Mech, 23:215–221, 1955
described a convention for standardizing the attachment of frames on links of a serial linkage
common convention for attaching reference frames on links of a serial manipulator and computing the transformations between frames
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105. Denavit-Hartenberg
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106. Denavit-Hartenberg notice the form of the rotation component
this does not look like it can represent arbitrary rotations
can the DH convention actually describe every physically possible link configuration?
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107. Denavit-Hartenberg
yes, but we must choose the orientation and position of the frames in a certain way
(DH1)
(DH2)
claim: if DH1 and DH2 are true then there exists unique numbers
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108. Denavit-Hartenberg
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109. Denavit-Hartenberg
proof: on blackboard in class
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110. Day 08
Denavit-Hartenberg
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111. Denavit-Hartenberg Forward Kinematics
RPP cylindrical manipulator
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112. Denavit-Hartenberg Forward Kinematics
How do we place the frames?
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113. Step 1: Choose the z-axis for each frame
recall the DH transformation matrix
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114. Step 1: Choose the z-axis for each frame
axis of actuation for joint i+1
link i
link i
link i+1
link i+1
joint i+1
joint i+1
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115. Step 1: Choose the z-axis for each frame
Warning: the picture is deceiving. We do not yet know the origin of the frames; all we know at this point is that each zi points along a joint axis
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116. Step 2: Establish frame {0}
place the origin o0 anywhere on z0
often the choice of location is obvious
choose x0 and y0 so that {0} is right-handed
often the choice of directions is obvious
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117. Step 2: Establish frame {0}
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118. Step 3: Iteratively construct {1}, {2}, ... {n-1}
using frame {i-1} construct frame {i}
DH1: xi is perpendicular to zi-1
DH2: xi intersects zi-1
3 cases to consider depending on the relationship between zi-1 and zi
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119. Step 3: Iteratively construct {1}, {2}, ... {n-1}
Case 1
zi-1 and zi are not coplanar (skew)
ai angle from zi-1 to zi measured about xi
shortest line between
and
(out of page)
point of intersection
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120. Step 3: Iteratively construct {1}, {2}, ... {n-1}
Case 2
zi-1 and zi are parallel ( ai = 0 )
notice that this choice results in di = 0
point of intersection
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121. Step 3: Iteratively construct {1}, {2}, ... {n-1}
Case 3
zi-1 and zi intersect ( ai = 0 )
point of intersection
(out of page)
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122. Step 3: Iteratively construct {1}, {2}, ... {n-1}
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123. Step 3: Iteratively construct {1}, {2}, ... {n-1}
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124. Step 4: Place the end effector frame
“sliding”
“normal”
“approach”
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125. Step 4: Place the end effector frame
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126. Step 5: Find the DH parameters
ai : distance between zi-1 and zi measured along xi
ai : angle from zi-1 and zi measured about xi
di : distance between oi-1 to the intersection of xi and zi-1 measured along zi-1
qi : angle from xi-1 and xi measured about zi-1
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127. Step 5: Find the DH parameters
Link ai di qi 1 d1
q1* 2
-90
d2* 3
d3*
* joint variable
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128. More Denavit-Hartenberg Examples
Day 09
More Denavit-Hartenberg Examples
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129. Step 5: Find the DH parameters
Link ai di qi 1 d1
q1* 2
-90
d2* 3
d3*
* joint variable
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130. Step 6: Compute the transformation
once the DH parameters are known, it is easy to construct the overall transformation
Link ai di qi 1 d1
q1* 2
-90
d2* 3
d3*
* joint variable
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131. Step 6: Compute the transformation
Link ai di qi 1 d1
q1* 2
-90
d2* 3
d3*
* joint variable
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132. Step 6: Compute the transformation
Link ai di qi 1 d1
q1* 2
-90
d2* 3
d3*
* joint variable
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133. Step 6: Compute the transformation
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134. Spherical Wrist
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135. Spherical Wrist
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136. Spherical Wrist: Step 1
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137. Spherical Wrist: Step 2
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138. Spherical Wrist: Step 2
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139. Spherical Wrist: Step 4
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140. Step 5: DH Parameters Link ai di qi 4 -90 q4* 5 90 q5* 6 d6 q6*
-90
q4* 5 90
q5* 6 d6
q6*
* joint variable
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141. Step 6: Compute the transformation
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142. RPP + Spherical Wrist
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143. RPP + Spherical Wrist
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144. Stanford Manipulator + Spherical Wrist
Link ai di qi 1
-90
q1* 2 90 d2
q2* 3
d3* 4
q4* 5
q5* 6 d6
q6*
* joint variable
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145. SCARA + 1DOF Wrist Link ai di qi 1 a1 d1 q1* 2 a2 180 q2* 3 d3* 4 d4d1
q1* 2 a2
180
q2* 3
d3* 4 d4
q4*
* joint variable
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